On the second largest eigenvalue of some Cayley graphs of the symmetric group

Abstract

Let $$S_n$$ S n and $$A_{n}$$ A n denote the symmetric and alternating group on the set $$\{1,\ldots ,n\},$$ { 1 , … , n } , respectively. In this paper we are interested in the second largest eigenvalue $$\lambda _{2}(\Gamma )$$ λ 2 ( Γ ) of the Cayley graph $$\Gamma =\mathrm{Cay}(G,H)$$ Γ = Cay ( G , H ) over $$G=S_{n}$$ G = S n or $$A_{n}$$ A n for certain connecting sets H. Let $$1<k\le n$$ 1 < k ≤ n and denote the set of all k-cycles in $$S_{n}$$ S n by C(n, k). For $$H=C(n,n)$$ H = C ( n , n ) we prove that $$\lambda _{2}(\Gamma )=(n-2)!$$ λ 2 ( Γ ) = ( n - 2 ) ! (when n is even) and $$\lambda _{2}(\Gamma )=2(n-3)!$$ λ 2 ( Γ ) = 2 ( n - 3 ) ! (when n is odd). Further, for $$H=C(n,n-1)$$ H = C ( n , n - 1 ) we have $$\lambda _{2}(\Gamma )=3(n-3)(n-5)!$$ λ 2 ( Γ ) = 3 ( n - 3 ) ( n - 5 ) ! (when n is even) and $$\lambda _{2}(\Gamma )=2(n-2)(n-5) !$$ λ 2 ( Γ ) = 2 ( n - 2 ) ( n - 5 ) ! (when n is odd). The case $$H=C(n,3)$$ H = C ( n , 3 ) has been considered in Huang and Huang (J Algebraic Combin 50:99–111, 2019). Let $$1\le r<k<n$$ 1 ≤ r < k < n and let $$C(n,k;r) \subseteq C(n,k)$$ C ( n , k ; r ) ⊆ C ( n , k ) be set of all k-cycles in $$S_{n}$$ S n which move all the points in the set $$\{1,2,\ldots ,r\}.$$ { 1 , 2 , … , r } . That is to say, $$g=(i_{1},i_{2},\ldots ,i_{k})(i_{k+1})\dots (i_{n})\in C(n,k;r)$$ g = ( i 1 , i 2 , … , i k ) ( i k + 1 ) ⋯ ( i n ) ∈ C ( n , k ; r ) if and only if $$\{1,2,\ldots ,r\}\subset \{i_{1},i_{2},\ldots ,i_{k}\}.$$ { 1 , 2 , … , r } ⊂ { i 1 , i 2 , … , i k } . Our main result concerns $$\lambda _{2}(\Gamma )$$ λ 2 ( Γ ) , where $$\Gamma =\mathrm{Cay}(G,H)$$ Γ = Cay ( G , H ) with $$H=C(n,k;r)$$ H = C ( n , k ; r ) with $$1\le r<k<n$$ 1 ≤ r < k < n when $$G=S_{n}$$ G = S n if k is even and $$G=A_{n}$$ G = A n if k is odd. Here we observe that $$\begin{aligned} \lambda _{2}(\Gamma )\ge (k-2)! {n-r \atopwithdelims ()k-r} \frac{1}{n-r} \big ((k-1)(n-k) - \frac{(k-r-1)(k-r)}{n-r-1}\big ). \end{aligned}$$ λ 2 ( Γ ) ≥ ( k - 2 ) ! n - r k - r 1 n - r ( ( k - 1 ) ( n - k ) - ( k - r - 1 ) ( k - r ) n - r - 1 ) . We prove that this bound is attained in the special case $$k=r+1$$ k = r + 1 , giving $$\lambda _{2}(\Gamma )=r!(n-r-1)$$ λ 2 ( Γ ) = r ! ( n - r - 1 ) . The cases with $$H=C(n,3;1)$$ H = C ( n , 3 ; 1 ) and $$H=C(n,3;2)$$ H = C ( n , 3 ; 2 ) were considered earlier in [6].

The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles

“…Therefore, the second largest eigenvalue of Cay(S n , C(n, I)) is attained uniquely by (1 n ). Moreover, the multiplicity of this eigenvalue is equal to the square of the dimension of ρ In (19) we saw that λ I (n−1,1) > λ I (2,1 n−2 ) whenever I ⊆ {2, 3 . .…”

“…, n} we obtain further the exact value of this eigenvalue together with its multiplicity (Theorems 3. [19]. A summary of our main results can be found in Table 1, where the third column shows all possible partitions of n which achieve the strictly second largest eigenvalue and the last column indicates the multiplicity of this eigenvalue.…”

“…This method was further used by Huang and Huang [10] to determine the second largest eigenvalues of the Cayley graphs on the alternating group A n with connection sets {(1 2 i), (1 i 2) | 3 ≤ i ≤ n}, {(1 i j), (1 j i) | 2 ≤ i < j ≤ n} and {(i j k), (i k j) | 1 ≤ i < j < k ≤ n}, respectively. With the help of the representation theory of symmetric groups, Siemons and Zalesski [19] determined the second largest eigenvalue of Cay(G, C(n, k)), where G = S n or A n , and k = n or n − 1, with C(n, k) the set of all k-cycles in S n . In [15, Theorem 1.2], Maleki and Razafimahatratra proved that if n − k ≥ 2 is relatively small compared to n then the second largest eigenvalue of Cay(S n , C(n, k)) is achieved by the standard representation of S n .…”

Aldous' spectral gap property for normal Cayley graphs on symmetric groups

“…With the help of the representation theory of symmetric groups, Siemons and Zalesski [37] determined the second largest eigenvalue of Cay(G, C(n, k)), where G = S n or A n , k = n or n − 1, and C(n, k) is the set of all k-cycles in S n .…”